Global smooth solutions of compressible Navier–Stokes equations with degenerate viscosity and vacuum

We study free boundary problem of the one dimensional compressible isentropic Navier–Stokes equations with density–dependent viscosity when the initial density connects to the vacuum states continuously and is either of compact or infinite support. Precisely, the pressure and the viscosity coefficient are assumed to be proportional to ρ^γ and ρ^θ respectively, where ρ is the density, and γ and θ are positive constants. We prove the global existence of smooth solutions with large initial data when θ > 0 and γ ≥ 1 + θ. Since the power θ of the previous results on this topic does not exceed 2, the result of this paper fills at least the gap for large θ. The result includes also the case of the infinite support of the initial density, which just corresponds to the one when 0 < θ ≤ 1. Notice that two key estimates of the proof are the uniform lower bound of the density and the uniform L^∞ bound of the velocity with respect to the construction of the approximate solutions. In contrast to the traditional techniques relying on weighted energy estimates, they are proved independently by the comparison principle and the maximal principle, respectively. Moreover, we obtain some results on regularity up to boundary and uniqueness of solutions. The results of this paper cover some important models, for instance, the viscous Saint–Venant model for the motion of shallow water, i.e., θ = 1 and γ = 2. © 2026 Elsevier Inc.